Was Theodred older or younger than Eomer?

Answer 1

Older, though the exact number of years is not given in the book.

In Appendix A of The Lord of the Rings, in the section titled "The House of Eorl: The Third Line," we learn this:

"Thus it came about that he [Eomund, father of Eomer and Eowyn, and husband of Theodwyn, Theoden's sister] was slain in 3002; for he pursued a small band to the borders of the Emyn Muil, and was there surprised by a strong force that lay in wait in the rocks.

Not long after, Theodwyn took sick and died to the great grief of the king....He [Theoden] had only one child of his own, Theodred his son, then twenty-four years old; for the queen Elfhild had died in childbirth, and Theoden did not wed again."

Earlier in this passage, we learn that Eomer was born in 2991 T.A., or about eleven years before Theodwyn's death, depending on how long "shortly after is." As Theodred was twenty-four in 3002-3, and Eomer was eleven to twelve years old, the conclusion is reached that Theodred was about 11-12 years older than Eomer.